Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities

拟变分不等式（QVIs）经常出现在非光滑和非线性现象导致复杂状态依赖约束的应用中。它们可以用来描述，例如，超导体的磁化，扭转中的热塑性效应，颗粒材料的行为，或细菌在竞争中的趋化行为。从数学上讲，这些问题的解决方案并不是唯一的，它们对输入量（数据、控制等）的依赖是不平滑的。本课题致力于椭圆型和抛物型qvi的最优控制问题的分析和数值求解。研究工作组织如下：(a)首先开发针对障碍型或梯度型约束的基于函数空间的QVIs解算法。特别地，我们的目标是路径跟踪半光滑牛顿格式，它具有快速的局部网格无关收敛性。(b)然后重点讨论了潜在QVIs的增强解理论。更具体地说，最小和最大解的性质将与相关的（微分）稳定性和数值近似方案一起研究。(c)然后，在两个要求越来越高的研究步骤中，将推导出所关心的QVIs的最优控制问题的平稳条件。这些优化问题属于函数空间中具有平衡约束的数学规划的范畴。在技术术语中，在我们的平稳性考虑中，将采用两种平滑方法，一种使用Moreau-Yosida技术，另一种依赖于修改底层微分算子的技术。(d)最后，讨论了所考虑的MPEC数值解的无束隐式规划方法。这些还包括松弛和路径跟踪技术，以及先进的离散化方案。项目工作中的分析和数值进展将针对原型应用进行验证。这些特别涉及到超导体的磁化、扭转中的热塑性效应、颗粒材料的行为以及细菌在竞争中的趋化行为。

Quasi-variational inequalities (QVIs) often arise in applications where non-smooth and nonlinear phenomena lead to complex state-dependent constraints. They can be used to describe, for instance, the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, or the chemotactic behavior of bacteria in competition. Mathematically, the solutions to these problems are not unique and their dependence on input quantities (data, controls, etc.) is non-smooth.This project is devoted to analyzing and numerically solving optimal control problems associated with elliptic and parabolic QVIs. The research work is organized as follows:(a) It starts with the development of function-space based solution algorithms for QVIs tailored to constraints of obstacle- or gradient-type. In particular, we aim at path-following semi smooth Newton schemes which exhibit fast local mesh-independent convergence.(b) Then it focuses on an enhanced solution theory for the underlying QVIs. More specifically, properties of the minimal and maximal solutions will be studied along with associated (differential) stability and numerical approximation schemes.(c) Then, in a two progressively more demanding research steps, stationary conditions for optimal control problems for the QVIs of interest will be derived. These optimization problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. In technical terms, in our stationarity considerations two smoothing approaches will be pursued, one utilizing a Moreau-Yosida technique and the the other one relying on a technique modifying the underlying differential operators. (d) Finally, bundle-free implicit programming methods for the numerical solution of the MPEC under consideration are pursued. These also involve relaxation and path-following techniques, and advanced discretization schemes. The analytical as well as numerical advance in the project work will be validated against prototypical applications. These involve in particular the magnetization of superconductors, thermoplastic effects in torsion, the behavior of granular material, and the chemotactic behavior of bacteria in competition.